Second-order functional-difference equations. I: Method of the Riemann-Hilbert problem on Riemann surfaces

An analytical method for scalar second-order functional-difference equations with meromorphic periodic coefficients is proposed. The technique involves reformulating the equation as a vector functional-difference equation of the first order and reducing it to a scalar Riemann-Hilbert problem for two finite segments on a hyperelliptic surface. The final step of the procedure is solution of the classical Jacobi's inversion problem. The method is illustrated by solving in closed form a second-order functional-difference equation when the corresponding surface is a torus. The solution is constructed in terms of elliptic functions.